applied sciences Article The Theoretical Study of an Interconnected Suspension System for a Formula Student Car Andrei-Cristian Pridie and Csaba Antonya * Department of Automotive and Transport Engineering, Transilvania University of Brasov, RO-500036 Brasov, Romania; [email protected] * Correspondence: [email protected]; Tel.: +40-268-418-967 Abstract: When it comes to racing applications, the primary engineering goal is to increase the performance envelope of the vehicle for a given set of tires. To achieve this goal, it is necessary to maximize the normal loads on the wheels while at the same time minimizing the tire load variation. The purpose of this paper is to present a mathematical model for a Formula Student car in order to study if performance gains are achieved by replacing the traditional passive suspension with a hydraulically interconnected suspension system. To have a complete picture of the advantages and disadvantages of each system, two vibrating models with 7 degrees of freedom were created in order to simulate the motion response of a Formula Student car to realistic excitations. Two particular interpretations of the results were chosen as important performance indicators. The first one is given by the pitch stability of the chassis relative to the road, which can be linked with a decrease in downforce load variation. The second one is the ability of the wheel to follow the road profile as closely as possible, which can be directly correlated with the amount of mechanical grip of the vehicle. The simulation results indicate that the hydraulically interconnected suspension system offers better results for both proposed cases but at the expense of the roll stability of the vehicle. Citation: Pridie, A.-C.; Antonya, C. Keywords: hydraulic suspension; vibrating model; formula student; interconnected suspension The Theoretical Study of an Interconnected Suspension System for a Formula Student Car. Appl. Sci. 2021, 11, 5507. https://doi.org/ 1. Introduction 10.3390/app11125507 The suspension system of a vehicle is represented by the totality of rigid and elastic links between the chassis of a vehicle and the wheels, which allows for the relative motion Academic Editor: Alessandro Gasparetto between the two. In commercial vehicle designs, the main goal of the suspension system is to provide to the driver and passengers of the vehicle as much comfort as possible. Received: 10 May 2021 This is usually achieved by limiting the amplitude of vertical accelerations experienced Accepted: 7 June 2021 Published: 14 June 2021 by the human occupants. In racing applications, however, the role of the suspension is to increase the performance envelope of the vehicle in order to have better dynamic Publisher’s Note: MDPI stays neutral results. The performance envelope of the vehicle is essentially a graphical representation with regard to jurisdictional claims in of the maximum achievable lateral and longitudinal accelerations for a given velocity. By published maps and institutional affil- studying the performance envelope of a racing car, we can see that the maximum lateral iations. and braking acceleration achievable are a function of the sum of all tire friction ellipses, while the forward acceleration of the vehicle is engine-limited. Another conclusion we can draw from the performance envelope is the fact that an increase of velocity will also increase the maximum achievable lateral and braking accelerations. This is a result of the fact that additional normal loads are added to the tires by the downforce generated at the higher Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. traveling velocity. Given the fact that any increase of the performance envelope translates This article is an open access article directly to better dynamic performances, two targets are set for the design engineer of distributed under the terms and a race car. conditions of the Creative Commons The first target is the increase of the mechanical grip of the vehicle by improving the Attribution (CC BY) license (https:// exploitation of the tire properties. The current engineering consensus when it comes to tire creativecommons.org/licenses/by/ usage is that a minimization of the tire load variation [1] can be correlated with an overall 4.0/). increase in the grip provided. Appl. Sci. 2021, 11, 5507. https://doi.org/10.3390/app11125507 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 5507 2 of 27 The second target of the design engineer must be the increase of the amount of downforce generated by the aerodynamic elements mounted on the vehicle. On a modern racing car, the majority of downforce (60–70%) is generated by the underbody of the car because it is usually designed to function as a venturi tunnel. However, the downforce generated by flow of air under the body is very sensitive to changes of the pitch angle. Studies done on IndyCar vehicles demonstrated that up to 50% can be lost if the pitch of the car changes just 1.3◦ [1]. Therefore, minimizing the pitch oscillation of the car is necessary to increase the aerodynamic grip. The vibration behavior of the vehicle describes how the wheels of the vehicle (the unsprung mass) oscillate relative to the chassis of the vehicle (the sprung mass). When considering a conventional vehicle with four wheels, this relative oscillation of the un- sprung mass can be categorized into four natural modes of vibration. The bounce mode occurs when all the wheels of the vehicle oscillate in phase. The roll mode occurs when the wheels on the same side are in phase while the ones on the opposite side are in counter phase. The pitch mode occurs when the wheels at one end are in phase and the ones at the opposite ends are in counter phase. Finally, twist or wrap mode occurs when the diagonally opposed wheels are in phase [2]. All other combinations of wheel displacement can be described by a combination of these 4 modes. By looking at Figure1, we can also see that the only nonplanar mode of vibration is the wrap mode. This kind of wheel oscillation relative to the sprung mass can only be excited by the road surface; therefore, any kind of stiffening for this mode will only increase the load variance of the wheel [3], with detrimental effects on the overall mechanical grip of the vehicle. Figure 1. Modes of vibration. (a) Bounce, (b) roll, (c) pitch, and (d) wrap. Appl. Sci. 2021, 11, 5507 3 of 27 The discussion about the natural modes of vibration allows us to distinguish some shortcomings of the conventional suspension system. By “conventional suspension sys- tem”, we understand a suspension system in which each individual wheel is connected to the chassis with a combination of elastic and dissipation elements. This suspension layout creates a coupling of the natural modes of vibration because the same elastic and dissipation elements must deal with all possible oscillations of the wheels. As such, the natural frequencies of the system are a function of the stiffness of the springs, the distance from the center of gravity of the vehicle (CG) and the wheels, and the mass properties of the sprung mass. Naturally, the ability to choose each natural mode of vibration independently would significantly improve the performance of the suspension system by eliminating the necessity to compromise. Some improvements have been made, however, with the addition of the anti-roll bar (ARB). By interconnecting the wheels at the same end of the vehicle with another elastic element that will work in series with the wheel springs, the roll stiffness of the vehicle can be modified. The addition of this device, however, will inevitably increase the wrap stiffness of the vehicle. Another disadvantage of this system is the fact that, although radial dampers exist, they are usually not added in order to decrease the complexity and save weight. Therefore, although the roll stiffness of the vehicle is decoupled, the roll damping of the vehicle is still coupled because it is done by the wheel damper. Another modification of the conventional suspension layout done to have more flexibility over the stiffnesses of the individual vibration modes is the addition of the third spring (also known as the heave spring). The purpose of this heave spring (which is paired with its own damper) is to decouple both the stiffness and dissipation of the bounce mode of vibration [4]. While this arrangement allows for the independent choice of stiffness for the bounce and roll modes of vibration, the pitch stiffness is coupled with the bounce modes because only the heave spring is deformed when either the bounce or pitch modes are excited. Another disadvantage of this arrangement is the fact that the roll and wrap modes of vibration are still coupled, therefore increasing the stiffness of the wrap mode of vibration. These deficiencies of the traditional suspension layout created the necessity for alter- native designs to solve the coupling problem. One possible solution to these challenges is the implementation of a hydraulically interconnected suspension system. Although not new, these solutions have not been explored to their full potential due to the overall higher level of complexity as well as higher expenses necessary for implementing and maintaining such a system. However, given the theoretical advantages that such a system can provide, a more in-depth analysis of such a system is required. The implementation of hydraulic interconnected suspension systems for motorsport applications has been tried before with some promising results. A short review of literature on hydraulically interconnected suspension systems shows that the first one was proposed in 1927 [5] by J. B. Hawley. This patent application proposed the replacement of all dampers with double-acting cylinders that could be interconnected based on the sought-after effect.